However, in this paper we restrict ourselves only to the liminf-estimates, which is sufficient to obtain convergence of the solutions, an approach also taken in [1]}, [2]}, [3]}. Namely, by a lower semicontinuity and compactness argument, Theorem REF  implies the convergence of both the solutions and the free energies \(\mathcal {F}_n\) , if the initial data are well prepared.
